I am in the process of ordering some book on the Langlands program, and learning more about it. In the mean time, I have a question which is easy to the experts, but being a beginner, I am not a 100% sure, so I could use some confirmation (or the opposite!).
In a trace formula, the geometric side is the trace of the matrix, which is the sum of the diagonal entries, while the spectral side is the sum of the eigenvalues. That these 2 sums are equal is the content of the trace formula.
If one were to label one side of the Langlands program as geometric, and the other as spectral, which side would that be?
If I were to guess, I would say that perhaps the Galois side is the spectral side, while the automorphic side is the geometric side, because I think that the Galois side is the side containing the information that we would like to obtain, so it is the spectral side, while the automorphic side is supposedly easier to calculate. Am I right, or do I have the analogy backwards?
I know that this question is philosophical. I asked it first on StackExchange, but neptun suggested that I try to ask it here. It may generate some interesting answer(s), which is what I am hoping for. In any case, I will learn something. I got inspired to ask the question after reading some notes by David Ben-Zvi (which I may have misunderstood, in which case I apologize), but I still know very little about the Langlands program.
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